#7626286

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3518	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,4,3)(7,13)(11,14)(15,19)(18,20)(21,22,27,25)(23,31,32,28)(24,30,26,29)(33,35,34,36);(5,6,10,8)(7,11,13,14)(9,16,17,12)(15,20,19,18)(22,25)(23,32)(29,30)(33,34);(1,3,4,2)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23,27,32)(22,28,25,31)(24,33,26,34)(29,35,30,36);(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24,27,26)(22,29,25,30)(23,33,32,34)(28,35,31,36);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,25,27,22)(23,31,32,28)(24,30,26,29)(33,36,34,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,26,27,24)(22,30,25,29)(23,34,32,33)(28,36,31,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533482);